Block #212,310

TWNLength 9★☆☆☆☆

Bi-Twin Chain · Discovered 10/16/2013, 5:17:37 AM · Difficulty 9.9186 · 6,582,420 confirmations

TWN

Bi-Twin Chain

Interleaved pairs of primes that differ by 2, forming twin prime pairs at each level.

Block Header

Block Hash

ca7e4153bc01231c430acb0056f48c2158f4b710d8465a8fdb267cd69b10c919

View on Chainz ↗

Height

#212,310

Difficulty

9.918643

Transactions

Size

4.90 KB

Version

Bits

09eb2c2f

Nonce

1,164,839,359

Timestamp

10/16/2013, 5:17:37 AM

Confirmations

6,582,420

Mined by

⛏️ V=R^!@VAenoL7BkxsuvjRARFVmi35JCjsDtRssSvW

Merkle Root

c36b384931c7d0eed33de560070fc4cf4c2e66ba7c1b53a71af576158c1348b0

Transactions (1)

Coinbasec36b384931c7d0…f576158c1348b0

1 in → 143 out10.1000 XPM4.81 KB

AenoL7Bk…DtRssSvW APw6W8tk…RgVVbThY AGB4r7xv…BBsSHL1w AYdkvntT…TCE18Zvb AMse1voq…GntYtmba

Prime Chain Origin

This is the prime chain origin stored in the block header. It is a composite number (not prime itself) — it equals the first prime in the chain multiplied by a primorial. The origin anchors the entire chain to this specific block.

3.662 × 10⁹³(94-digit number)

36623534060988636819…44100507342002251599

Discovered Prime Numbers

Lower: 2^k × origin − 1 | Upper: 2^k × origin + 1

These are the actual prime numbers discovered by this block, computed using the verified Primecoin formula. Each number has been independently confirmed to pass the Fermat primality test. Use the FactorDB links to verify any number independently.

Level 0 — Twin Prime Pair (origin ± 1)

origin − 1

3.662 × 10⁹³(94-digit number)

36623534060988636819…44100507342002251599

Verify on FactorDB ↗Wolfram Alpha ↗

origin + 1

3.662 × 10⁹³(94-digit number)

36623534060988636819…44100507342002251601

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: origin + 1 − origin − 1 = 2 (twin primes ✓)

Level 1 — Twin Prime Pair (2^1 × origin ± 1)

2^1 × origin − 1

7.324 × 10⁹³(94-digit number)

73247068121977273638…88201014684004503199

Verify on FactorDB ↗Wolfram Alpha ↗

2^1 × origin + 1

7.324 × 10⁹³(94-digit number)

73247068121977273638…88201014684004503201

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^1 × origin + 1 − 2^1 × origin − 1 = 2 (twin primes ✓)

Level 2 — Twin Prime Pair (2^2 × origin ± 1)

2^2 × origin − 1

1.464 × 10⁹⁴(95-digit number)

14649413624395454727…76402029368009006399

Verify on FactorDB ↗Wolfram Alpha ↗

2^2 × origin + 1

1.464 × 10⁹⁴(95-digit number)

14649413624395454727…76402029368009006401

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^2 × origin + 1 − 2^2 × origin − 1 = 2 (twin primes ✓)

Level 3 — Twin Prime Pair (2^3 × origin ± 1)

2^3 × origin − 1

2.929 × 10⁹⁴(95-digit number)

29298827248790909455…52804058736018012799

Verify on FactorDB ↗Wolfram Alpha ↗

2^3 × origin + 1

2.929 × 10⁹⁴(95-digit number)

29298827248790909455…52804058736018012801

Verify on FactorDB ↗Wolfram Alpha ↗

Difference: 2^3 × origin + 1 − 2^3 × origin − 1 = 2 (twin primes ✓)

Level 4 — Twin Prime Pair (2^4 × origin ± 1)

2^4 × origin − 1

5.859 × 10⁹⁴(95-digit number)

58597654497581818910…05608117472036025599

Verify on FactorDB ↗Wolfram Alpha ↗

What this block proved

The miner who found this block proved the existence of 9 consecutive prime numbers forming a Bi-Twin Chain. The prime chain origin — the large number shown above — anchors the chain and is divisible by a primorial (the product of small primes), cryptographically tying these prime numbers to this specific block.

★☆☆☆☆

Rarity

CommonChain length 9

Found in most blocks. The baseline for Primecoin's proof-of-work.

How Primecoin's Proof-of-Work Constructs These Primes

Primecoin stores a value called the prime chain origin in each block. The miner found a large integer such that when divided by a primorial (the product of small primes: 2 × 3 × 5 × 7 × …), the result is the first prime in the chain. The origin is deliberately divisible by this primorial — that divisibility is part of the proof.

Prime Chain Origin = First Prime × Primorial (2·3·5·7·11·…)

Source: Primecoin prime.cpp — CheckPrimeProofOfWork()

This is why the origin has many small prime factors — those factors are the primorial divisor. The chain then extends from the first prime using the TWN formula:

TWN: twin pairs (p, p+2) where p = origin/primorial − 1 and p+2 = origin/primorial + 1

Cross-reference on Chainz Explorer